Question

Compton used photons of wavelength \(0.0711 \mathrm{nm}\) a) What is the wavelength of the photons scattered at \(\theta=180 .^{\circ} ?\) b) What is energy of these photons? c) If the target were a proton and not an electron, how would your answer to part (a) change?

Short answer

Answer: The scattered wavelength with an electron as a target is approximately 0.0748 nm, and the energy of the scattered photons is approximately 2.66 x 10^-15 J. If the target were a proton instead of an electron, the scattered wavelength would be approximately 0.0711 nm, not significantly different from the initial wavelength.

Step-by-step solution

Calculate the scattered wavelength using the Compton scattering formula

The Compton scattering formula for the scattered wavelength (λ') is given by: $$ \lambda' = \lambda + \frac{h}{m_e c}(1-\cos{\theta}) \, $$ where λ is the initial wavelength, h is the Planck constant, \(m_e\) is the electron mass, c is the speed of light, and θ is the scattering angle. We are given the initial wavelength as 0.0711 nm and the angle as 180°. Converting the angle from degrees to radians: $$ \theta_{radians} = \theta_{degrees} \cdot \frac{\pi}{180}= 180 \cdot \frac{\pi}{180} = \pi \, radians $$ Now plug in the given values and constants to find the scattered wavelength: $$ \lambda' = 0.0711 + \frac{6.626 \times 10^{-34}}{9.109 \times 10^{-31} \cdot 2.998 \times 10^8}(1-\cos{\pi}) \, nm $$ Calculating the scattered wavelength: $$ \lambda' \approx 0.0748 \, nm $$

Find the energy of the scattered photons

We can use the Planck-Einstein relation to find the energy of the scattered photons: $$ E = \frac{hc}{\lambda'} $$ Plug in the values: $$ E = \frac{6.626 \times 10^{-34} \cdot 2.998 \times 10^8}{0.0748 \times 10^{-9}} \, J $$ Calculating the energy: $$ E \approx 2.66 \times 10^{-15} \, J $$

Compare the scattered wavelength with a proton and an electron

If the target were a proton instead of an electron, the Compton scattering formula would become: $$ \lambda'_{proton} = \lambda + \frac{h}{m_p c}(1-\cos{\theta}) \, $$ where \(m_p\) is the mass of a proton. It is important to note that the mass of a proton is much more than that of an electron (approximately 1836 times). The rest of the constants and given values remain the same. As a result, the additional term in the formula \((\frac{h}{m_p c}(1-\cos{\theta}))\) would be much smaller compared to the case of an electron, making the scattered wavelength nearly equal to the initial wavelength (λ). And so, if the target were a proton instead of an electron, the scattered wavelength would be approximately 0.0711nm, not significantly different from the initial wavelength.

Key concepts

Compton Effect

The Compton effect is a quantum mechanical phenomenon that occurs when a photon (a particle of light) scatters off a target particle, typically an electron. When the photon collides with the electron, it transfers some of its energy and momentum to the electron, resulting in a photon with lower energy and hence a longer wavelength.

Discovered by Arthur Compton in 1923, it was a landmark observation because it demonstrated the particle nature of light, supporting the notion that light can behave as both a wave and a particle, known as wave-particle duality. Importantly, the larger the angle of scattering, the greater the energy transfer from the photon to the electron, meaning the final wavelength of the photon increases with the scattering angle.

Photon Energy Calculation

The energy of a photon is calculated using the Planck-Einstein relation, which links the energy (E) of a photon to its frequency (u) or, equivalently, to its wavelength (\r(\rllambda)) since the speed of light (c) is a constant. The relation is given by: \[\r( E = hu = \frac{hc}{\r(\rllambda))}) \]

Here, \r(h) is Planck's constant, and the frequency and the wavelength are inversely related. Shorter wavelengths correspond to higher energies and longer wavelengths to lower energies. This relation enables us to understand and quantify the energy involved in phenomena such as Compton scattering, where the energy of scattered photons can be determined from their measured wavelengths.

Compton Wavelength Shift

During Compton scattering, the wavelength shift of the photon, which is the difference between the initial and final wavelengths, is a direct measure of how much energy is transferred to the electron. This wavelength shift is described by the Compton equation: \[\r(\rllambda') = \r(\rllambda) + \frac{h}{m_e c}(1-\r(\rcos{\r(\rtheta)}))}) \]

Here, \r(\rllambda') is the scattered wavelength, \r(\rllambda) is the original wavelength, \r(\rtheta) is the angle of scattering, \r(h) is the Planck constant, \r(m_e) is the mass of the electron, and \r(c) is the speed of light. The term \r(\frac{h}{m_e c}) is known as the 'Compton wavelength' of the electron. It's a constant that underscores the quantum mechanical nature of the scattering process and it's crucial in determining the wavelength shift based on the scattering angle.

Planck-Einstein Relation

The Planck-Einstein relation establishes a critical connection between the energy of a photon and its wavelength or frequency. It's expressed as: \[\r( E = \frac{hc}{\r(\rllambda))} \]

In this formula, \r(E) represents the energy of the photon, \r(h) is Planck's constant (6.626×10^-34 Js), \r(c) is the speed of light in a vacuum (approximately 3×10^8 m/s), and \r(\rllambda) is the wavelength of the photon. This equation is frequently used in quantum mechanics and is essential for calculations involving photon energies, such as during the analysis of spectroscopic data, photoelectric effect experiments, and, of course, understanding the nuances of Compton scattering.